Solving two-dimensional linear partial differential equations based on Chebyshev neural network with extreme learning machine algorithm

Abstract

Purpose

This paper aims to develop a novel algorithm and apply it to solve two-dimensional linear partial differential equations (PDEs). The proposed method is based on Chebyshev neural network and extreme learning machine (ELM) called Chebyshev extreme learning machine (Ch-ELM) method.

Design/methodology/approach

The network used in the proposed method is a single hidden layer feedforward neural network. The Kronecker product of two Chebyshev polynomials is used as basis function. The weights from the input layer to the hidden layer are fixed value 1. The weights from the hidden layer to the output layer can be obtained by using ELM algorithm to solve the linear equations established by PDEs and its definite conditions.

Findings

To verify the effectiveness of the proposed method, two-dimensional linear PDEs are selected and its numerical solutions are obtained by using the proposed method. The effectiveness of the proposed method is illustrated by comparing with the analytical solutions, and its superiority is illustrated by comparing with other existing algorithms.

Originality/value

Ch-ELM algorithm for solving two-dimensional linear PDEs is proposed. The algorithm has fast execution speed and high numerical accuracy.

Citation

Liu, M., Hou, M., Wang, J. and Cheng, Y. (2021), "Solving two-dimensional linear partial differential equations based on Chebyshev neural network with extreme learning machine algorithm", Engineering Computations, Vol. 38 No. 2, pp. 874-894. https://doi.org/10.1108/EC-08-2019-0387